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•<>•<>• GENERALIZED PROFILE NUMBERS Department SHMUEL ZAKS of Computer Science, Technion, (Submitted December 1981) Haifa, Israel INTRODUCTION A family of binary trees {T^} is studied in [2]. The numbers p(n, k) of internal nodes on level k in Tn (the root is considered to be on level 0) are called profile numbers, and they "enjoy a number of features that are strikingly similar to properties of binomial coefficients" (from [2]). We extend the results in [2] to t-ary trees. DISCUSSION We discuss t-ary trees (see Knuth [1]). A t-ary tree either consists of a single root, or a root that has t ordered sons, each being a root of another £-ary tree. 58 [Feb. GENERALIZED PROFILE NUMBERS Let p t (n5 k) denote the number of internal nodes at level k in the tree T*. The numbers p t (n9 k) satisfy the recurrence relation pt (n + 1, k + 1) = (t - l)pt (w, k) +tpt(n9 k - 1) (1) together with the boundary conditions pt (w, 0) = 1 Ptd, 1) = 1 p (n, 1) = t (2) for n > 1 p. (1, k) = 0 for fc> 1. The corresponding trees and sequences for the case of binary trees (t = 2) is studied in [2]. Thus9 Tn and p(n5 k) in [2] are denoted here by T% and p2(ns k), respectively. We first show that pt (n, k) = tk-n E <3> (* ~ ^'(i)' 0^i<2n-k where n ^ 1 9 k ^ 0 9 and the / . J? s are the binomial coefficients. Note that when k < n we have p (n,fc)= tfee The expression in (3) is easily shown to satisfy the boundary conditions (2) e To continue3 we induct on n (and arbitrary k); using (1) and the inductive hypothesis, we get + 1, k + 1) = (£ - l)pt(n,fc)+ tp t (n 5 k - 1) pt(n 0<7:<2n-k -**-* 0<i<2n-fe+l E 0<£<2n-&:+l = **-» + **-» (*-D<(i:1) + *fc-+*kV x) ( * - ^ ( " t 1 ) = **-» 0<i<2n-k+l E (*-i)*(;) 0<£<2n-k+l ^-l)i{n+il) E 0<i<2n-k +l and this establishes (3)* Using (3) 5 we get pt(n9 19537 k + 1) =tpt(n9 k) - tk~n +1 (t - D2"-*"1^- _ * + x) (4) 59 GENERALIZED PROFILE NUMBERS and pt(n k) + tk-n~Ht + 1, k) = pt(n, (k\)-^-<:'„)] ^ D2n~k - where n > 1 and k ^ 0. Let x* be the number of internal nodes in Tn , namely < = L Pt(». k>- (6) 0 < k < In Using (3) , changing the order of summation, and applying the binomial theorem results in m «.• • -t ^ - '• «> Note that, by their definition, the numbers x* satisfy the recurrence relation x\ ^+i = 2 = (2t ~ l^xt + 2 for i > 0, (8) which also implies (7). Let in denote the internal path length (see [1]) of Tn , namely £ n = E kpt(n, (9) k). 0 < k < In The numbers ln also satisfy the recurrence relation l\ =1 £* + 1 = (It - l)li + (3t - 1 ) ^ + 1 for i > 0. (10) Using (9) and (3), or solving (10) with the use of (7), one gets (* - I ) 2 The average level e% of a node in T% is thus given by %nlx\ , and satisfies < ~ H Z I n + 0(1). (12) The results in (1), (2), (3), (4), (5), (7), and (11) are extensions of (1), (3), Theorems 1, 2a, 2b, 3, and 4 of [2], respectively. 60 [Feb. GENERALIZED PROFILE NUMBERS If we denote F (x * > y) = X Ptt(n> k)xny*9 then, using (1) and (2), we get Ft(x9 y) = *- ^ (1 - x)(1 - txy + xy - . (13) txy1) Equations (1) and (7), for the case t = 2 9 were noted in [2] to be similar to the recurrence relation and the summation formula 0<fc<n The binomial coefficients also satisfy E<-»'U)-°Using (3) 5 one can show that the same identity holds for any t and n; namelys X) (-l)*Pt(n,fc)= 0. (14) 0^k<2n REFERENCES 1. D. E. Knuth. Algorithms. 2* The Art New York? A. L. Rosenberg. 259-264. of Computer Programming. Vol, I: Fundamental Quarterly 17 (1979): Addison-Wesley9 1.968. "Profile Numbers." Fibonacci • 0404 1983] 61